Exam 40: One-Dimensional Quantum Mechanics

An electron is bound in an infinite well (a box) of width 0.10 nm. If the electron is initially in the n = 8 state and falls to the n = 7 state, find the wavelength of the emitted photon. (c = 3.00 × 10⁸ m/s, h = 6.626 × 10^-34 J ∙ s, m_el = 9.11 × 10^-31 kg)

A particle is confined to a one-dimensional box (an infinite well) on the x-axis between x = 0 and x = L. The potential height of the walls of the box is infinite. The normalized wave function of the particle, which is in the ground state, is given by ψ(x) = sin , with 0 ≤ x ≤ L. What is the maximum probability per unit length of finding the particle?

An electron with kinetic energy 2.80 eV encounters a potential barrier of height 4.70 eV. If the barrier width is 0.40 nm, what is the probability that the electron will tunnel through the barrier? (1 eV = 1.60 × 10^-19 J, m_el = 9.11 × 10^-31 kg, h = 1.055 × 10^-34 J ∙ s, h = 6.626 × 10^-34 J ∙ s)

An electron is confined in a harmonic oscillator potential well. What is the longest wavelength of light that the electron can absorb if the net force on the electron behaves as though it has a spring constant of 74 N/m? (m_el = 9.11 × 10^-31 kg, c = 3.00 × 10⁸ m/s, 1 eV = 1.60 × 10^-19 J, h = 1.055 × 10^-34 J ∙ s, h = 6.626 × 10^-34 J ∙ s)

A 3.10-eV electron is incident on a 0.40-nm barrier that is 5.67 eV high. What is the probability that this electron will tunnel through the barrier? (1 eV = 1.60 × 10^-19 J, m_el = 9.11 × 10^-31 kg, H = 1.055 × 10^-34 J ∙ s, h = 6.626 × 10^-34 J ∙ s)

The lowest energy level of a particle confined to a one-dimensional region of space (a box, or infinite well) with fixed length L is E₀. If an identical particle is confined to a similar region with fixed length L/6, what is the energy of the lowest energy level that the particles have in common? Express your answer in terms of E₀.

An electron is confined in a harmonic oscillator potential well. A photon is emitted when the electron undergoes a 3→1 quantum jump. What is the wavelength of the emission if the net force on the electron behaves as though it has a spring constant of 9.6 N/m? (m_el = 9.11 × 10^-31 kg, c = 3.00 × 10⁸ m/s, 1 eV = 1.60 × 10^-19 J, h = 1.055 × 10^-34 J ∙ s, h = 6.626 × 10^-34 J ∙ s)

You want to confine an electron in a box (an infinite well) so that its ground state energy is 5.0 × 10^-18 J. What should be the length of the box? (h = 6.626 × 10^-34 J ∙ s, m_el = 9.11 × 10^-31 kg)

A particle confined in a rigid one-dimensional box (an infinite well) of length 17.0 fm has an energy level E_n = 2.0 MeV and an adjacent energy level E_n+1 = 37.5 MeV. What is the value of the ground state energy? (1 eV = 1.60 × 10^-19 J)

An electron in an infinite potential well (a box) makes a transition from the n = 3 level to the ground state and in so doing emits a photon of wavelength 20.9 nm. (c = 3.00 × 10⁸ m/s, h = 6.626 × 10^-34J ∙ s, m_el = 9.11 × 10^-31 kg) (a) What is the width of this well? (b) What wavelength photon would be required to excite the electron from its original level to the next higher one?

An electron is in an infinite square well (a box) that is 8.9 nm wide. What is the ground state energy of the electron? (h = 6.626 × 10^-34 J ∙ s, m_el = 9.11 × 10^-31 kg, 1 eV = 1.60 × 10^-19)

Find the wavelength of the photon emitted during the transition from the second EXCITED state to the ground state in a harmonic oscillator with a classical frequency of 3.72 × 10¹³ Hz. (c = 3.00 × 10⁸ m/s, 1 eV = 1.60 × 10^-19 J, h = 1.055 × 10^-34 J ∙ s, h = 6.626 × 10^-34 J ∙ s)